Mathematica has nice features for display of various data structures, but it does its computations on very simple, often List-based structures. Many of the display features will interfere with computation. As a general rule (not 100%), any Mathematica function that ends in Form (e.g. MatrixForm, TableForm) is for display purposes only. When the front end "sees" MatrixForm, it creates a special "object" that makes the pretty display. If you try to do subsequent computations on that object, you will get unexpected results. Two of these *Form "functions" are useful for debugging: InputForm and FullForm. Also useful for debugging are various "functions" that prevent evaluation.

So, to debug your notebook, here's what I did...

For your first input, I copied the cell contents to my clipboard by hovering over it and choosing "copy" (which is just a feature of this website). I pasted directly into a new notebook and then surrounded it with Hold and FullForm like this:

Then I evaluated that, which produced the following:

Hold[Eigensystem[List[List[6, 2, 1], List[0, 4, 2], List[0, 0, 3]]]]

That looks fine. I did the same thing with your next cell:

and I got this:

Hold[MatrixForm[
  Inverse[Grid[List[List[1, -1, 1], List[0, 1, -2], List[0, 0, 1]], 
    Rule[ItemSize, List[Inherited, Inherited]]]]]]

And now we can see a problem. Instead of taking Inverse of a matrix, you're taking Inverse of a Grid expression, which doesn't make any sense. I don't know how you got a Grid expression in there -- you'll need to think back on your process to figure that out. But if we get rid of that Grid, it should work fine:

Inverse[List[List[1, -1, 1], List[0, 1, -2], List[0, 0, 1]]]
(* output is {{1, 1, 1}, {0, 1, 2}, {0, 0, 1}} *)

Of course, List[List[1, -1, 1], List[0, 1, -2], List[0, 0, 1]] is the fully expanded expression (what FullForm shows you). You could just as easily have used the "normal" input form (what InputForm shows you):

Inverse[{{1, -1, 1}, {0, 1, -2}, {0, 0, 1}}]

Continue this process for the rest of you notebook. You'll see that you have Grid littered throughout your code, and that is interfering with the matrix calculations.

Side note... ClearAll isn't used as a standalone command. You need to tell it which symbols you want to clear. One common usage is ClearAll["Global`*"]. In your case, I don't see that your code defines any symbols, so there is no need to use ClearAll.

You are using Grid to compose matices. Don't do that.

Thank you pardon me. What do you mean by use InputForm. What is a MatrixFormWrapper?

How do I get code in pastable for?

(1) Use InputForm to see what it is you are actually evaluating. You will find a MatrixForm wrapper that is getting in the way.

(2) In general to get assistance one should provide actual code in copy-pastable form.